Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 4.3.5.8. Let $\operatorname{\mathcal{C}}$ be a category containing an object $X$, which we also view as a vertex of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Specializing Example 4.3.5.7 (and invoking Example 4.3.1.9), we obtain canonical isomorphisms

\[ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/X} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/X} ) \quad \quad \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})_{X/} \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{X/} ). \]